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Physical metallurgy principles
Chapter 5 Dislocations and Plastic Deformation
Frank – read source* A dislocation generator
The dislocation segment xy may
move in plane ABCD under the
applied stress. Its ends, x and y,
however, are fixed.
Various stages in the generation
of a dislocation loop at a Frank-
Read source.
Nucleation of dislocations If dislocations are not formed by dislocation
generators (Frank-Read source), then they must be created by a nucleation process.
Nucleation process created in two ways:
Homogeneous nucleation: formed in a perfect lattice by the action of a simple stress, no agency other than stress being required.
Heterogeneous nucleation: the dislocations are formed with the help of defects present in the crystal, perhaps impurity particles.
Defects make the formation of dislocations easier by lowering the applied stress required to form dislocations.
That homogeneous nucleation of dislocation requires extremely high stresses.
If dislocations are not formed by Frank-Read sources, then they must be nucleated heterogeneously.
Bend gliding
The bending of crystals can be explained in terms of Frank-Read or other sources.
Elastic deformation: (before the yield point of crystal)
the stress distribution
section-cross theof inertia ofmoment the:
axis neutral thefrom measured distance verticalthe:
moment bending the:
I
y
MI
Myx
The shear stress component parallel to the slip plane.
The sense of the shear stress changes its sign as it crosses the neutral axis.
The shear stress is zero at the neutral axisand a maximum at the extreme ends of the slip plane.
upper: compressive stress; lower: tensile stress
The stress distribution on slip planes corresponding to
the elastic deformation.
positive edge : move toward the surface (high stress region) and disappear
negative edge : move toward the specimen’s neutral
axis(decreasing shear stress)
neutral axis : free of dislocations, not be stressed above the elastic limit, deformation will be elastic and not plastic
Rotational slip
Type of deformation due to dislocations: simple shear, bending, rotational slip.
Rotational slip required more than one set of dislocations(slip plane must contain more than one slip direction)
Ex. FCC , and HCP
An array of parallel screw dislocations A double array of screw dislocations
Critical resolved shear stress Critical resolved shear stress: the yield
stress for crystals of a shear stress resolved on the slip plane and in the slip direction.
plane slip on the resolved stressshear :
plane slip on the stress :
coscoscos
cos
cos
A
n
nA
n
n
sp
nA
sp
n
A
f
A
f
A
f
A
A
•When a crystal possesses several
crystallographically equivalent slip systems,
slip will start first on the system having the
highest resolved shear stress.
FCC slip systems
* Close-packed plane: {111}
* Close-packed directions:<110>
* Slip systems: 4 x 3=12
* Large number of equivalent slip system
a: several slip systems have nearly equal resolved shear stresses.
b: 1: only one slip plane
2: multiple glide on intersecting slip planes
3: decreasing the rate of increase of the dislocation density
HCP slip system
* Close-packed plane: basal plane (0001)
* Close-packed direction: <11-20>
Titanium (0001) 110
Titanium (10-10) 49
Beryllium (0001) 39
Zirconium (10-10) 6.2
Zinc, cadmium and magnesium possess both a low critical resolved
shear stress and a single primary slip plane (basal plane)
Just for Zn and Cd
BCC slip system
Close-packed direction: <111>
Slip plane (contains a close-packed <111> direction) : {110},{112},{123}
The lack of close-packed plane causes high critical resolved shear stress for slip.
Ex. Fe: 28 MPa
Cross slip Cross-slip: there are two or more slip planes with a
common slip direction occur in crystals.
Only screw dislocation can shift of the dislocation from one plane to another in the cross-slip.
Cross slip for total and extended dislocations
A. a total dislocation can readily cross-slip between a pair of octahedral planes.
B. ½ [-110]1/6[-211] +1/6[-12-1]
C. cross-slip of the leading partial , 1/6[-211] 1/6[-121]+1/6[-1-10]
D. cross-slip of the trailing partial, 1/6[-12-1]+1/6[-1-10]1/6[-21-1]
stair-rod
cross-slip for a total is much easier than extended s due to the formation of
stair-rod .
Work hardening
a point: elastic limit, begin to deform plastically and neck.
The true strength of the metal normally increases with increasing strain until it fractures.
strain gengineerin :
strain true:
stress gengineerin :
stress true:
)1ln(ln
)1(
0
t
t
t
t
l
lA
P
Geometrical instability
<u,
Due to large work hardening capacity,
instability cannot continue.
Localized strain at “weak line”
occurs regularly and repeatly.
>u,
work hardening rate decrease to
a point that instability once formed
continuous to develop.
40
41
42
43
44
Instability in tension
-Necking generally begins at max. load during the tensile deformation of a
ductile metal. An idea plastic material in which no strain hardening occurs
would become unstable in tension and begin to neck just as soon as
yielding took place.
-a real metal undergoes strain hardening, which tends to increase the
load-carrying capacity of the specimen as deformation increases. This
effect is opposed by the gradual decrease in the cross-sectional area of
the specimen as it elongates.
-Necking or localized deformation begins at max load, where the increase
in stress due to decrease in the cross-sectional area becomes greater
than the increase in the load-carrying ability due to strain hardening.
-The condition of instability leading to localized deformation is defined by
the condition dP=0.
45
46
t
t
t
t
tt
Necking criterion (Considère’s criterion)
47
Relationship between dislocation density and the stress
r is the measured dislocation density, k is a constant , and 0 is the stress obtain when is extrapolated to zero
The relationship in terms of the resolved stress on the active slip plane,
Taylor’s relationthe dislocation interaction must be overcome in order to allow the dislocations to continue to glide
k = amb
Observing dislocation in crystal by etching reagent, which forms an etch pit on the surface of a crystal at each point where a dislocation intersects the surface.
The velocity of a dislocation moving under a fixed applied stress = distance that a dislocation moved by the time
Strain and dislocation velocity
V=d/t , v: dislocation velocity ; d: dislocation motion distance ; t: time of application of the stress.
Edge dislocation would normally move 50 times faster than screw dislocation.
kTE
m
efv
Tv
scmv
ex
cm/
DD
v
vv
/
55.16
)(
1
ln :stressconstant
/101.13.5
65.2 :n velocitydislocatio
MPa 2.65 16.5,m MPa, 5.3D:crystal LiF .
sec 1 of
n velocitydislocatio a yields that stress:
stressshear : )(
n velocitydislocatio: lnln
-
•The movement of dislocation is not smooth and continuous,
but rather it occurs in steps.
•Thermal vibrations aid the applied stress to overcome these
these obstacles to dislocation motion.
obstaclean at n waitsdislocatio the timeaverage:
obstaclesbetween flight of time:
obstaclesbetween distance average the:
:n velocitydislocatio
w
f
wwf
t
t
l
t
l
tt
lv
V
nldensity n dislocatio:
V
xbnl
crystal theofheight :z ; :strain shear Az
Ab
passedn dislocatio offraction the:A
movesn dislocatio of distance:x tor;burger vec:
)/()/(:amount sheared
ex.
rate.strain applied and
ndislocatio of velocity ebetween th iprelationsh :equationOrowan
vbt
xb
t
xb
V
Ab
b
AAbxxb
rr
r
r
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